Average Error: 41.5 → 0.0
Time: 10.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.001473090169205698964430273356640554993646:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - 1 \cdot e^{-x}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.001473090169205698964430273356640554993646:\\
\;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\

\mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - 1 \cdot e^{-x}}\\

\end{array}
double f(double x) {
        double r105645 = x;
        double r105646 = exp(r105645);
        double r105647 = 1.0;
        double r105648 = r105646 - r105647;
        double r105649 = r105646 / r105648;
        return r105649;
}
↓
double f(double x) {
        double r105650 = x;
        double r105651 = -0.001473090169205699;
        bool r105652 = r105650 <= r105651;
        double r105653 = exp(r105650);
        double r105654 = r105650 + r105650;
        double r105655 = exp(r105654);
        double r105656 = 1.0;
        double r105657 = r105656 * r105656;
        double r105658 = r105655 - r105657;
        double r105659 = r105653 / r105658;
        double r105660 = r105653 + r105656;
        double r105661 = r105659 * r105660;
        double r105662 = 0.0017165907253591187;
        bool r105663 = r105650 <= r105662;
        double r105664 = 0.08333333333333333;
        double r105665 = 1.0;
        double r105666 = r105665 / r105650;
        double r105667 = fma(r105664, r105650, r105666);
        double r105668 = 0.5;
        double r105669 = r105667 + r105668;
        double r105670 = -r105650;
        double r105671 = exp(r105670);
        double r105672 = r105656 * r105671;
        double r105673 = r105665 - r105672;
        double r105674 = r105665 / r105673;
        double r105675 = r105663 ? r105669 : r105674;
        double r105676 = r105652 ? r105661 : r105675;
        return r105676;
}

Error

Bits error versus x

Target

Original41.5
Target41.0
Herbie0.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.001473090169205699

    1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied flip--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}\]

    4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{e^{x + x} - 1 \cdot 1}} \cdot \left(e^{x} + 1\right)\]

    if -0.001473090169205699 < x < 0.0017165907253591187

    1. Initial program 62.4

      \[\frac{e^{x}}{e^{x} - 1}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}}\]

    if 0.0017165907253591187 < x

    1. Initial program 38.8

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied clear-num38.8

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Simplified0.5

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied div-inv0.5

      \[\leadsto \frac{1}{1 - \color{blue}{1 \cdot \frac{1}{e^{x}}}}\]

    7. Simplified0.4

      \[\leadsto \frac{1}{1 - 1 \cdot \color{blue}{e^{-x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001473090169205698964430273356640554993646:\\ \;\;\;\;\frac{e^{x}}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)\\ \mathbf{elif}\;x \le 0.001716590725359118732584740030233660945669:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - 1 \cdot e^{-x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))